where “*” denotes values at horizon crossing, and \(n_*\)  is the statistical distribution function of inflaton fluctuations; these may remain in vacuum (\(n_*=0\) ), they may termalize due to the interactions (\(n_* = (1 - e^{H_*/T_*})^{-1}\) ), or be in some other intermediate state. On the other hand, the function \(G[Q_*]\)  encodes what is called the “growing mode”, due to the coupling of inflaton and radiation fluctuations. At linear order, it depends on the derivative \(d \ln \Upsilon / d \ln T\)  at horizon crossing, such that when this is positive it does amplify the primordial spectrum, whereas it has the opposite effect for a negative derivative. When \(d \ln \Upsilon /d \ln T=0\)  the perturbation equations can be solved analytically at first order in the slow-roll parameters [1]}, obtaining Eq. (REF ) with \(G[Q_*]=1\) ; otherwise the determination of \(G[Q_*]\)  requires the numerical integration of the system. This is important because \(G[Q_*]\)  and its derivative will directly enter in the expressions for the spectral index \(n_s\)  and the tensor-to-scalar ratio \(r\) .
